Whakaoti i te {r}{32x+3y=5}{3x+2y=70}

$$\left. \begin{array} { r } { 32 x + 3 y = 5 } \\ { 3 x + 2 y = 70 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{40}{11} = -3\frac{7}{11} \approx -3.636363636$$

$$y = \frac{445}{11} = 40\frac{5}{11} \approx 40.454545455$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Hei whakaoti i ētahi whārite takirua mā te whakakapinga, me whakaoti tētahi whārite i te tuatahi mō tētahi o ngā taurangi. Ka whakakapi i te otinga mō taua taurangi ki tērā o ngā whārite.

$$32x+3y=5,3x+2y=70$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te x mā te wehe i te x i te taha mauī o te tohu ōrite.

$$32x+3y=5$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$32x=-3y+5$$

Whakawehea ngā taha e rua ki te 32.

$$x=\frac{1}{32}\left(-3y+5\right)$$

Whakareatia \frac{1}{32} ki te -3y+5.

$$x=-\frac{3}{32}y+\frac{5}{32}$$

Whakakapia te \frac{-3y+5}{32} mō te x ki tērā atu whārite, 3x+2y=70.

$$3\left(-\frac{3}{32}y+\frac{5}{32}\right)+2y=70$$

Whakareatia 3 ki te \frac{-3y+5}{32}.

$$-\frac{9}{32}y+\frac{15}{32}+2y=70$$

Tāpiri -\frac{9y}{32} ki te 2y.

$$\frac{55}{32}y+\frac{15}{32}=70$$

Me tango \frac{15}{32} mai i ngā taha e rua o te whārite.

$$\frac{55}{32}y=\frac{2225}{32}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{55}{32}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{445}{11}$$

Whakaurua te \frac{445}{11} mō y ki x=-\frac{3}{32}y+\frac{5}{32}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{32}\times \frac{445}{11}+\frac{5}{32}$$

Whakareatia -\frac{3}{32} ki te \frac{445}{11} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{1335}{352}+\frac{5}{32}$$

Tāpiri \frac{5}{32} ki te -\frac{1335}{352} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{40}{11}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{40}{11},y=\frac{445}{11}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$32x+3y=5,3x+2y=70$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}32&3\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\70\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}32&3\\3&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}32&3\\3&2\end{matrix}\right))\left(\begin{matrix}32&3\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}32&3\\3&2\end{matrix}\right))\left(\begin{matrix}5\\70\end{matrix}\right)$$

Ko te hua o tētahi poukapa me te kōaro ko te poukapa tuakiri.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}32&3\\3&2\end{matrix}\right))\left(\begin{matrix}5\\70\end{matrix}\right)$$

Whakareatia ngā poukapa kei te taha mauī o te tohu ōrite.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}32&3\\3&2\end{matrix}\right))\left(\begin{matrix}5\\70\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right) te poukapa kōaro, nō reira ka taea te tuhi anō te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{32\times 2-3\times 3}&-\frac{3}{32\times 2-3\times 3}\\-\frac{3}{32\times 2-3\times 3}&\frac{32}{32\times 2-3\times 3}\end{matrix}\right)\left(\begin{matrix}5\\70\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{55}&-\frac{3}{55}\\-\frac{3}{55}&\frac{32}{55}\end{matrix}\right)\left(\begin{matrix}5\\70\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{55}\times 5-\frac{3}{55}\times 70\\-\frac{3}{55}\times 5+\frac{32}{55}\times 70\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{40}{11}\\\frac{445}{11}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{40}{11},y=\frac{445}{11}$$

Ngā Upane e Whakamahi i te Tangohanga

Hei whakaoti mā te tangohanga, ko ngā tau whakarea o tētahi o ngā taurangi me mātua ōrite i ngā whārite e rua kia whakakorehia ai te taurangi ina tangohia tētahi whārite mai i tētahi atu.

$$32x+3y=5,3x+2y=70$$

Kia ōrite ai a 32x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 32.

$$3\times 32x+3\times 3y=3\times 5,32\times 3x+32\times 2y=32\times 70$$

Whakarūnātia.

$$96x+9y=15,96x+64y=2240$$

Me tango 96x+64y=2240 mai i 96x+9y=15 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$96x-96x+9y-64y=15-2240$$

Tāpiri 96x ki te -96x. Ka whakakore atu ngā kupu 96x me -96x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$9y-64y=15-2240$$

Tāpiri 9y ki te -64y.

$$-55y=15-2240$$

Tāpiri 15 ki te -2240.

$$-55y=-2225$$

Whakawehea ngā taha e rua ki te -55.

$$y=\frac{445}{11}$$

Whakaurua te \frac{445}{11} mō y ki 3x+2y=70. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+2\times \frac{445}{11}=70$$

Whakareatia 2 ki te \frac{445}{11}.

$$3x+\frac{890}{11}=70$$

Me tango \frac{890}{11} mai i ngā taha e rua o te whārite.

$$3x=-\frac{120}{11}$$

Whakawehea ngā taha e rua ki te 3.

$$x=-\frac{40}{11}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{40}{11},y=\frac{445}{11}$$